Distribution of the Number of Times M/M/2/N Queuing System Reaches its Capacity in Time t under Catastrophic Effects

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1 International Journal of Computational Science and Mathematics. ISS Volume 3, umber 4 (), pp International Research Publication House Distribution of the umber of Times M/M// Queuing System Reaches its Capacity in Time t under Catastrophic Effects Dhanesh Garg and.k. Jain Research Scholar, Department of Statistics and Operational Research, Kurukshetra University, Kurukshetra, Haryana-369, India dhaneshgargind@gmail.com Professor, Department of Statistics and Operational Research, Kurukshetra University, Kurukshetra, Haryana-369, India nkjain@kuk.ac.in Abstract This paper presents the distribution of the number of times that a limited capacity to servers Markovian queue subjected to catastrophes reaches its capacity in time t. The occurrence of a catastrophe makes the system empty instantaneously. After the occurrence of a catastrophe both the servers become ready to serve the ne customers. The afore said distribution is obtained as a marginal distribution of the joint distribution of the number of customers in the system at time t and the number of times system reaches its capacity in time t. Some measures of effectiveness are obtained explicitly ith catastrophic effects. Keyords: Catastrophes, homogeneous servers, markovian queue, p.g.f. Introduction In the last one decade a lot of ork has been done by various authors taking into consideration the concept of catastrophe Kumar et. al. [5 studied the transient behaviour of the M/M/ queue ith catastrophes. Crescenzo et. al. [ made a continuous approximation of M/M/ queue ith catastrophe. Chao et. al [9 obtained the transient analysis of immigration birth-death processes ith total catastrophe. In all the mentioned studies the researchers have obtained the state probabilities in one ay or the other and have computed various measures of performance. In this paper e have explicitly obtained the distribution of the number of times the system prone

2 39 Dhanesh Garg and.k. Jain to catastrophes reaches its capacity in time t. various other measures of performance are obtained explicitly. Model Description ith Catastrophe We consider an M/M// queueing system having to homogenous servers ith FCFS discipline subjected to catastrophe. The customers arrive at a counter in accordance ith a Poisson process ith mean arrival rate λ>. Each server serves one customer at a time if available. The service time distribution of a customer is negative exponential ith mean rate μ>. The queuing process starts at time zero ith zero state of the system. Catastrophes occur according to Poisson process ith mean rate ξ only hen the system is not empty. The occurrence of catastrophe makes the queuing system empty instantaneously, and simultaneously both the servers become ready to accept the ne customers. We define P (t) = Prob. [X(t) = m, Y(t) = n, n m, n [ X (t) = the number of times the system reaches its capacity in time t. Y (t) = the number of customers in the system at time t. The marginal probabilities are P., n() t = Pm, n() t m = and P ( t) = P ( t) m,. m, n n = Differential-Difference Equations Governing the System d P,() t m = ( λ+ P m, () t + μ P m, () t + ξ P m,. () t ; n =, m dt d P, () t mn = ( λ+ μ+ P mn, () t + λ P mn, () t μ P mn, () t ; n, m + + < < dt d P,() t m = ( λ+ μ+ P m, () t + λ P m, () t + μ P m,() t ; n =, m dt d P, () t m = ( μ+ P m, () t + λ P m, () t ; n, m = dt [ [3 [4 [5

3 Distribution of the umber of Times M/M// Queuing System Reaches 39 Taking Laplace transform of the equations [-[5.r.t t e have λ ξ μ ξ ( s+ + ) P,( s) = + P,( s) + P,. ( s); n=, m= λ ξ μ ξ ( s+ + ) Pm, () s = Pm, () s + Pm,. (); s n=, m> m, m, m, ( s + λ+ μ+ P ( s) = λp ( s) + μp ( s); n= s + λ+ μ+ ξ P s = λp s + μp s n< ( ) mn, ( ) mn, ( ) mn, + ( ); s + μ+ ξ P s = λp s n= ( ) m, ( ) m, ( ); [6 [7 [8 [9 [ Since P,() = st mn, = mn, P () s e P () t dt Define the probability generating functions by Pn( xs, ) Pm, n( sx ) m = m= H ( xys, ; ) = Pn( xsy, ) n= P. xs Pm,. sx (, ) = ( ) m m= n [ [ [3 Multiplying equation [6 to [ by x m, summing over the ranges of m and using [, e have λ ξ μ ξ ( s+ + ) P( x, s) = + P ( x, s) + P. ( x, s); n= λ μ ξ λ μ ( s ) P( x, s) = P( x, s) + P( x, s); n= λ μ ξ λ μ ( s+ + + ) Pn( xs, ) = Pn ( xs, ) + Pn+ ( xs, ); n=,3,..., [4 [5 [6 ( μ (, ) λ (, ); s + + P x s = xp x s n= [7

4 39 Dhanesh Garg and.k. Jain Multiplying equation [4 to [7 by y n, summing over the ranges of n and using [, 3, e have on simplification: sy( x)( y) + ys [ + ξ( y) s( y)[ μ+ ys ( + λ+ P ( xs, ) + λ ( ) P ( xs, ) λx y + sy ( x ) + s+ μ+ ξ Hxys (, ;) = ss [( + y+ ( y)( λy μ) [8 Since λx = s + μ + ξ P ( xs, ) P( xs, ) The zeros of the denominator in [8 are given by ( s+ λ+ μ+ ± ( s+ λ+ μ+ 4 λμ αi () s =, i =,. λ [9 The existence of H(, xys ;) is only possible if numerator vanishes for α and α the to zeros of the denominator. This ill give rise to equations, solving them e have: P ( x, s) = P ( x, s) = A x s (, ) Axs (, ) A (., s) Axs (, ) λx A( xs, ) = μλ { V ( + ) V ( )} ξγ { V ( ) V ( ) } + ( x) sv { () V ( )} ξ { V ( ) γ V ( ) } s + μ+ ξ [ A (., s ) = μ ( s + ξ )( s + λ ) + μξ V () s+ λ+ ξ ( x) V() { V ( ) γ V ( )} { V ( ) γ V ( ) } λ Axs (,) = μλ s x + λ{ V ( + ) γ V ( + ) } + ( s+ { V ( + ) γ V ( )} ( s+ λ+ { V ( ) γ V ( ) } s + μ+ ξ V( r) = α ( s) α ( s); r =,,,... r r μ ( ) αα = = γ λ say [ [

5 Distribution of the umber of Times M/M// Queuing System Reaches 393 If e rite ( s + y+ ( y)( λy μ) =λ( α y)( α y) n y = y n= ( α ) α ( / α ) n y = y n= ( α ) α ( / α ) [8 Yields ys y s y ys P xs sy( x)( y) + P sy ( x ) + s+ μ+ ξ xs sv λ () [ + ξ( ) ( )[ μ+ ( + λ+ (, ) + λ ( ) (, ) + λx y n n y y α α n= α n= α [ o Pn( xs, ) is the coefficient of y n in [. Comparing the coefficients of y n on both sides of [, e have: n γ Pn (,) xs = { μγ s Tn () + ss ( + λ+ ( Tn)} P(,) xsξtn ( ) ( s+ Vn (), n sv λ () [3 T (n) = V (n+) - γ V (n) Since P( xs, ) [3 gives is given by [, so Pn( xs, ) are explicitly knon. For setting x=, { μγ Tn ( ) + ( s+ λ+ Tn ( ) }{ sv [ () V ( ) ξ T ( ) } { λv() + λt( ) + ( s+ λ+ T( ) } γ () = ( + () ξ ( ) sv λ () [4 n P, n s s Vn Tn Applying the Leibniz differentiation theorem to [, setting x= and dividing both sides by m!, e have:

6 394 Dhanesh Garg and.k. Jain P m, λγ ( s+ μ + V() γ λ T( + ) ( s λ + μ + { sv() ( s+ V( ) + ξγ V( ) } [( s+ ξγ ) T( ) λv( ) m { λγ [( s+ μ + V() + μt ( ) + γ T ( )[( s+ λ + + μ( s+ λ + γ λ V ( + ) } () s = m m sγ ( s+ μ+ { λv() + λt( ) + ( s λ T( ) } m [5 Whence form [3 { μγ λ ξ } γ () = s T( n) + s( s+ + ) T( n) () ( + ) () ( ), m sv λ () [6 n Pmn, s Pm, s s ξ V n ξt n We kno that i ( j+ ) j+ i a j+ j j j i j ji i ( ab / ) =,( + ) = ( + ) + ( + ) ( ) = ( ) j ( i ) i ( i ) C a b a a b b a b and a b C a b b i= i= Using these Identities in [4 and [6, e have j+ l j+ l+ r () s = γ ν ( ) λ γ { μγ j= l= h= k= r= l h r j+ l n l+ h ( j+ ) ( j l+ ) ( j+ kl), n P ( ) ( ) ( ) ( ) ([ g + j + h l n g + j + h l + n + R R γ [ R g + j + h l n R g + j + h l + n ) + ( s+ λ+ [ R + + ( ( k l ) ( g j h l n ) R γ [ R R ( g + j + h l + n ) ( g + j + h l n + ) ( g + j + h l + n ) i= i + i (i + ) γ ( λ) [( s + μ+ λ+ + ν ξ + γ ( λ) [( s + μ+ λ+ + ν s i= i= i + ( i+ ) (i + ) i + i (i + ) γ ( λ) [( s + μ+ λ+ + ν n i+ n n i+ n ( ni ) (in + ) ( ni ) (in + ) + γ ( λ) [( s+ μ+ λ+ + ν 3[ γ ( λ) [( s+ μ+ λ+ + ν λ i= i= )} { } { }, Pmn() s n γλ ( ) [ + μγ R ( + ) R γ [ R n R n ( s λ [ R n R n γ [ R n ( n ) R = [7

7 Distribution of the umber of Times M/M// Queuing System Reaches 395 m j++ i l h jl i++ l h+ p+ q m m+ j j+ i+ k j+ i+ l h j l j+ l+ q+ r () i= j= l= h= k= p= q= r= i j k l h p p r λ μγ + μ+ ξ + λ+ ξ n i+ n ( ni ) (in + ) 3 ( ) [( ) i + γ λ s μ λ ξ ν ( i+ ) (i + 3) i= ξs γ ( λ) [( s+ μ+ λ+ + ν λ n i+ n i= ( ni ) (in + ) γ ( λ) [( s+ μ+ λ+ + ν i= mil + p mil ( ) ( ) ( ) ( ) ( ) + pq l m + jiq mil p mil p g jih p g jih p ( s ) + ( s ) + [ R R i + n i n + i (i + ) ( ni) (i n+ ) ξλ s γ ( λ) [( s+ μ+ λ+ + ν + γ ( λ) [( s+ μ+ λ+ + ν i= i= [8 R = αγ ν = s + μ + λ + ξ λμ and g = l + k + h + h, ( ) 8 [ ( ) We are no in a position to complete the solution for the joint distribution of X (t) and Y (t). Taking the Inverse Laplace transform of [7 and [8, using the tables [3, 8, e have: j+ l j+ l+ r P () t =, n { I I j+ l l+ h ( j+ ) ( j l+ ) n ( j+ kl) () λ γ μγ [ ( g++ j h l n ) ( g++++ j h l n ) j= l= h= k= r= l h r t ( k l ) ( λ+ μ+ t ( t) [ I( g++ j h l n) I( g+++ j h l n) } e { [ I( g++ j h l n) I( g+++ j h l n) γ [ I( g+++ j h l n ) I( g+++ j h l n ) } ( k l )! γ + t ( λ+ μ+ ( ) ( λ+ μ+ t ( λμe ) d γ (i + ) I( i + ) t e + ξ [ γ (i + ) I( i + ) i= i= n n ( ) n n+ ( λ+ μ+ [ γ (i + ) I( i + ) + λ [ γ (i n+ ) I( i+ n ) 3[ γ (i n+ ) I( i+ n ) ( λμ ) e d i= i= i= P () t mn, m j++ i l h jl i++ l h+ p+ q = () i= j= l= h= k= p= q= r= i j k l h p p r m m j j i k j i l h j l j l q r t mikl + ( u) { ( A n) I + ( A+ n+ 3) I ( mil + p) ( mil + pq ) l ( m+ jiqn ) λ μγ μγ ( An ) μ ( An + + 3) ( m+ i k l )! + ( A n+ 3) I + γ ( A+ n) I + γ [ μγ ( A+ n+ ) I γ( A n+ ) I ( An + 3) ( An + ) ( An ++ ) ( An + ) mikl + ( λ+ μ+ u ( u) ( An + ) } λμ ( An + ) ( An ++ ) ( m+ i k l )! ( λ+ μ+ ( An + 3) γ ( An + ) γ ( An ++ ) γ ( An + ) λμ + μ( A n+ ) I ( u) u e du+ ( A n+ ) I ( A+ n+ ) I + ( A n+ 3) I + ( A+ n) I + [( A+ n+ ) I ( + )( A n+ ) I ( u) u e [9 u du d t n n ( ) n+ n ( λ+ μ+ ( ) ( 3) (i + 3) [ ( ) (i n+ ) ( ) (i n+ ) ( ) i= i= i= ξ t γ i + I λ γ i n+ I γ i n+ I λμ e d

8 396 Dhanesh Garg and.k. Jain t n ( ) n+ ( λ+ μ+ ( ) (i + ) ( ) (i n+ ) ( ) i= i= ξλ γ i + I + γ i n+ I λμ e d [3 A = g + j + i + p, I ( αt) I and α = λμ ν ν umber of Times the System Reaches its Capacity in Time t under Catastrophic Effects Setting y= in [8, e have: H ( x,; s) ( s ξ P.( x, s) λ( x) P = + + ( x, s) [3 Using [ in [3, e have: λ( x)[( s+ ( s+ λ) + μξ V() Hx (,;) s= ( s+. + ξpxs (,) λsxs [( + μ+ { λt ( + ) + ( s+ ( T) ( s+ λ+ ( T)} + ( x){ V() + T ( ) + λ ( s++ λ T ( )} [3 Setting x= in [3, e have: [( s+ ( s+ λ) + μξ V() = + s[ λv() + λt( ) + ( s+ λ+ T( ) [33 P,. () s ( s ξ ) Differentiating [3, m times.r.t. x and dividing both sides by m! and setting x=, e have: λv()[( s+ ( s+ λ) + μξ[ λ T( + ) + λ( s+ T( ) λ( s+ λ+ T( ) ( s+ μ+ m { λv() λt( ) ( s λ T( )} Pm,. s m m m+ () =, s ( s+ μ+ [ λv() + λt( ) + ( s+ λ+ T( ) [34 Expanding the right hand sides of [33 and [34, e have:

9 Distribution of the umber of Times M/M// Queuing System Reaches 397 j j k k j+ r j= l= h= k= r= k l h r jk k j k+ l+ h ( j+ ) ( k+ ) ( j+ k+ ) ( kl) P,. () s = s () λ γ ( s+ λ+ jk k j ( g+ j) ( g+ j+ ) ( g+ j+ r) ( g+ j+ r+ ) k+ l+ h μγ [ R R γ [ R R μξs ( ) j + j= l= h= k= r= k j k k j+ r ( j+ ) ( k+ ) ( j++ k ) ( k l ) ( g+++ j r ) ( g+++ j r ) λ γ ( s+ λ+ [ R R l h r [35 P m,. m j++ i m i jk l i++ l p+ q+ h m m+ j j+ k j+ i+ mi () s = () i j k l p i= j= l= k= p= q= h= r= mil + pq mil + pq (m+ jiq ) l+ p ( mikl ) ( mil + p+ ) l j k j+ i+ l+ r λ γ μ ( s+ λ+ ( s+ μ+ h q r ( ) ( ) ( ) ( 3) ( ) ( ) [ B + p + B + p + [ B + p + + R R R R B + γ p + + μξ( s [ R B + r + R B + r [36 B = (g + j + i) Taking the Inverse Laplace transforms of [35 and [36, e have t jk k j k l k++ l h j j k k j+ r ( j+ ) ( k+ ) ( j+ k+ ) ( t) P,. () t = ( ) λ γ j= l= h= k= r= k l h r ( k l )! γ { μ[( g + j ) I( g+ j ) ( g + j + ) I( g++ j ) [( g + j + r ) I( g++ j r ) ( g + j + r + ) I( g+++ j r ) jk k j ( λ+ μ+ u k+ l+ h j+ j k k j+ r ( j+ ) ( λμuu ) e du dμξ ( ) j= l= h= k= r= k l h r t k l ( k+ ) ( j+ k+ ) ( t ) λ γ ( g+ j+ r+ ) I( g+ j+ r+ ) ( g+ j+ r+ ) I ( g+ j+ r+ ) ( kl)! ( λμuu ) e ( λ+ μ+ u du d m i l j k j+++ i l r p h q r m j++ i m i jk l i++ l p+ q+ h m m+ j j+ k j+ i+ Pm,.() t = ( ) i= j= l= k= p= q= h= r= i j k l λ γ μ t mil + p + p ( t) ( m+ i l p)! m+ i l pq m+ i l pq (m+ j i q) l [37

10 398 Dhanesh Garg and.k. Jain mikl ( u) [( B+ h+ p+ ) I( Bhp ++ + ) ( B+ h+ p+ ) I( Bhp ++ + ) γ[( B+ h+ p+ + ) ( m i k l )! I B h p I uu e u u ( λ+ μ+ u ( Bhp ) ( ) ( Bhp3) ( λμ ) + μξ ( + ( ν ) ( ) [( B+ h+ r+ ) I( Bhr +++ ) ( B+ h+ r+ ) ( ) λ+ μ+ ξ ν I ( Bhr) λμν ν e dν du d [ Measure of Effectiveness μ To measures of effectiveness of immediate interest are the expectation { ( )( ) X t } and σ the variance { ( ) X () t } of the distribution of the number of times the system reaches its capacity in time interval (, t. μ ( ) Expectation { ( ) X t } d μ X( s) ( ) H = ( x,; s) dx x= [39 Using [3 in [39 and after some algebraic calculation, e have: k j+ k+ l i+ l j+ k j+ k+ l l k ( ji k) + k( + ) ( jik) μxs ()( ) = ( ) γ ( s+ λ+ j= i= l= p= k= j l p i [4 ( ) ( ) ( ) ( ) ( 3) ( ) i l + p [ H + H s μ ξ λ R R λξ( λ μ) s [ R H + R H + Taking the inverse Laplace transform, e have: μ Xt () k j++ k l t j i k i+ l j+ k j+ k+ l l k ( ji k) + k( + ) ( t) ( ) = ( ) γ j= i= l= p= k= j l p i ( j i k)! ( u) i+ l p u λ[( H+ ) I ( H ) I + λξ( λ+ μ) [( H+ ) I ( H+ 3) I ( H+ ) ( H ) ( H+ ) ( H+ 3) ( i+ l p)! ( λ+ μ+ ν ( λμν ) ν e dν du d H= [3(j + i + k + l +) + [4 Variance { ( ) X () t σ } σ ( ) = Kt ( ) + μ ( )[ μ ( ) X() t X() t X() t [4

11 Distribution of the umber of Times M/M// Queuing System Reaches 399 K() t = m( m) P () t m= m,. Taking Laplace transform of K (t), e have: () = ( ) m,. () K s m m P s m= d Ks () = Hx (,;) s dx x= [43 Using [3 in [43 and after some algebraic calculation, e have i+ j+ i+ i j j+ i+ p+ Ks () = () λ i j k l p i j j= i= k= p= l= k+ l ( i+ j+ ) (k+ l+ + p( + )) ( ) 3 ( ) ( ) j + ( ) i s ξ s λ ξ [μξs 4 μ( μ s μξ( μ s ( s j + ( s λ i 3 ( C) ( C+ ) + + s s + + s R R [( μ λ ( μ (μ 4 λ λ( μ j l= p= r= γ j= i= k= i+ j+ i+ i j j+ i+ p+ j+ r+ () λ γ ( s+ i j k l p r k+ l ( i+ j+ 3) (k+ l+ + p( + )) ( j+ 3) ( s + λ+ [μξs + 4 μ( μ+ s + μξ( μ+ s + ( s+ ( s+ λ+ [( μ+ λ+ s ( ik) 3 ( j+ ) ( ik) 3 ( C+ ) ( C+ ( + )) + ( μ+ (μ+ 4 λ+ s + λ( μ+ s R R [44 i C = [i + j + (k + I + p + ) (+) Taking the Inverse Laplace transform, e have j t i k+ l i+ j+ i+ i j j+ i+ p+ ( i+ j+ ) (k+ l+ + p( + )) ( t) K(t)= ( ) λ γ j= i= k= p= l= i j k l p ( j + )! [μξ + 4 μ( μ + + μξ( μ + [( C) I( C ) ( C+ ) I( C+ ) ( i )!!! i u u u ( u) ( uν) ( uν) λμν ν ν + j! μ + λ + ξ + μ + ξ μ + λ + ξ ( i)!! t j i u u ( λ+ μ+ ( t) ( u) ( uν ) ( ) e d du d [( ) ( )( 4 ) u i j ( uν ) ( λ+ μ+ k+ l C I( C ) C I( C ) e d du d +! j= i= k= l= p= r= + λ( μ+ [( ) ( + ) ( λμν) ν ν ( ) i+ j+ i+ i j j+ i+ p+ j+ r ( t) ( u) [ i j k l p r ( j+ )! ( ik)! j+ t j+ ik ( i j 3) (k l p( )) λ γ μξ

12 4 Dhanesh Garg and.k. Jain u u u ( uν) ( uν) ( λ+ μ+ + 4 ( + ) + ( + ) [( ) ( C+ ) ( ) ( C+ + ) ( )! + + +! μ μ ξ μξ μ ξ C I C I λμν ν e dν t j+ ik u u u ( t) ( u) ( uν) ( uν) du d+ [( μ+ λ+ + ( μ+ (μ+ 4 λ+ + λ( μ+ ( j+ )! ( ik)!!! [( ) ( ) ( ) ( λ+ μ+ C + I( C+ ) C + + I( C+ + ) λμν ν e dν du d [45 References [ Chao, X, 995, A queueing netork model ith catastrophes and product form solution. O.R.Letters, 8, [ DiCrescenzo, A.Giorno, V and Ricciardi, L.M., 3, On the M/M/ queue ith catastrophes and its continuous approximation. Queueing systems,43, [3 Erdelyi, A, W.Magnus, F.Oberhettinger and F.G.Tricomi, 954, Table of Integral Transform, Vol., McGra-Hill Book Company, e York. [4 Kumar,B.K.,Arivudainambi,D.,,Transient Solution of an M/M/ queue ith catastrophes, Computers and math ith Applica. 4(), 33-4(8). [5 Kumar,B.Krishna,Madhesari,S.Pavai,,Transient behaviours of the M/M/ queue ith catastrophes, Statistica, anno LXII, n. [6 Singh, V.P., 97, To-server Markovian queues ith balking: Heterogeneous vs. homogeneous servers, Operations Research, Vol.9, [7 Watson, G.., Sc.D, F.R.S., 959, A Treatise on the Theory of Bessel Function, nd Edition, Cambridge Uni. Press. [8 Widder, D.V., 946, The Laplace Transform, Princeton Mathematical Series [9 Xiuli,Chao and Yuxi, Zheng,3,Transient analysis of immigration birthdeath processes ith total catastrophes, Probability in the Engineering and informational Sciences,7(), 83-6.